XXVIII. Exact Formulation of the General Principle of RelativityRelativity: The Special and General Theory
We are now in a position to replace the pro. visional formulation of the general principle of relativity given in Section XVIII by an exact formulation. The form there used, “All bodies of reference etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion,” cannot be maintained, because the use of rigid reference-bodies, in the sense of the method followed in the special theory of relativity, is in general not possible in space-time description. The Gauss co-ordinate system has to take the place of the body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: “All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”
We can state this general principle of relativity in still another form, which renders it yet more clearly intelligible than it is when in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz transformation, we replace the space-time variables , of a (Galileian) reference-body by the space-time variables , of a new reference-body . According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables , the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.
If we desire to adhere to our “old-time” three-dimensional view of things, then we can characterise the development which is being undergone by the fundamental idea of the general theory of relativity as follows: The special theory of relativity has reference to Galileian domains, i.e. to those in which no gravitational field exists. In this connection a Galileian reference-body serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of “isolated” material points holds relatively to it.
Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies also. A gravitational field of a special kind is then present with respect to these bodies (cf. Sections XX and XXIII).
In gravitational fields there are no such things as rigid bodies with Euclidean properties; thus the fictitious rigid body of reference is of no avail in the general theory of relativity. The motion of clocks is also influenced by gravitational fields, and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity.
For this reason non-rigid reference-bodies are used, which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion. Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time. We have to imagine each of these clocks fixed at a point on the non-rigid reference-body. These clocks satisfy only the one condition, that the “readings” which are observed simultaneously on adjacent clocks (in space) differ from each other by an indefinitely small amount. This non-rigid reference-body, which might appropriately be termed a “reference-mollusc”, is in the main equivalent to a Gaussian four-dimensional co-ordinate system chosen arbitrarily. That which gives the “mollusc” a certain comprehensibility as compared with the Gauss co-ordinate system is the (really unjustified) formal retention of the separate existence of the space co-ordinates as opposed to the time co-ordinate. Every point on the mollusc is treated as a space-point, and every material point which is at rest relatively to it as at rest, so long as the mollusc is considered as reference-body. The general principle of relativity requires that all these molluscs can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusc.
The great power possessed by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence of what we have seen above.
XXIX. The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity
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